There are several ways to interpret the homotopy hypothesis.
Strictly speaking, Grothendieck's homotopy hypothesis is not a theorem yet: Grothendieck stated it in a very precise way in the very first pages of Pursuing stacks, by defining explicitly a notion of weak $\infty$-groupoid and by associating functorially to any space $X$ its weak higher groupoid $\Pi_\infty(X)$. Grothendieck's notion of weak higher groupoid is very closely related to Batanin's (who restated the homotopy hypothesis using his formalism). In this sense the homotopy hypothesis might be seen as a syntactic statement. The theory of weak higher groupoids is based on a language (the one of globular objects=$\infty$-graphs), and the homotopy theory of weak higher groupoids has the nice property that, up to equivalence, every object is free in an appropriate sense (=being cofibrant).
This is illustrated as follows: consider your favourite $\infty$-graph and then the weak higher groupoid associated to it. Then, if you want to add a relation between two $n$-cells, you simply add freely a $(n+1)$-cell connecting them. This means that it is quite easy to actually define a weak higher groupoid out of any reasonable sentence formulated in the globular language, and that the resulting object will be characterized by a universal property both in a strict sense and in an homotopical sense. If you interpret such a language in the world of stacks of weak higher groupoids, this means that you will have a powerful way to understand geometric objects (having such a theory at hand is the main motivation of Grothendieck for the definition of weak higher groupoids).
The meaning of the homotopy hypothesis is related to the fact that the homotopy theory of CW-complexes is freely generated by the point under small homotopy colimits (and thus, any homotopy theory is canonically enriched in homotopy types). Therefore, the homotopy hypothesis means that the language of weak higher groupoids is not only convenient, but also universal: this is a very appropriate language for homotopy theory in general (in some sense, this is the language, or even better, the logic of CW-structures). I insist on language here because, after the work of Awodey, Voevodsky et al. on homotopy type theory, the homotopy hypothesis has even more meaning: the very syntax of (cofibrant) weak higher groupoids is very related to the formulation of what people from homotopy type theory call higher inductive types, and thus, a full understanding of the homotopy theory of higher groupoids (e.g. the homotopy hypothesis, seen as way to describe higher topoi as stacks of weak higher groupoids) would give powerful tools to this part of logic (and thus to people who create programming languages for proof assistants, for instance).
Now, the homotopy hypothesis, considered in the spirit of the nLab (and which should not be attributed to Grothendieck, then), is a theorem, if we define weak higher groupoids as Kan complexes (a theorem due to Milnor in the 1950's), but more importantly a principle to develop the theory of $(\infty,1)$-categories as the one of categories enriched in weak higher groupoids, without giving an explicit description of what such things are. But such a principle must be completed by another one: the language of ordinary category theory can be interpreted to speak of $(\infty,1)$-categories.
This latter principle can already be seen in the work of Dwyer and Kan on simplicial categories, but it has been promoted to another level with Joyal's insights: quasi-categories (aka Boardman and Vogt's weak Kan complexes) define a category in which the very syntax of category theory defines constructions which are homotopy invariant (i.e. define objects in the Joyal model category structure which can be described in terms of mapping spaces, homotopy (co)limits, etc), so that this syntactic interpretation can be translated in any other model of $(\infty,1)$-categories (as far as the translation is done through a zig-zag of Quillen equivalences, say). This gives very powerful tools to produce and characterize homotopy categories using the language of ordinary category theory only, and underlies all the beautiful work of Lurie and others.
In conclusion, we see that there are two problems here: having an appropriate synthetic language for the theory of $(\infty,1)$-categories (and then the language of ordinary category theory, together with the `easy' version of the homotopy hypothesis provide such a thing, through the non-trivial work of Joyal, Lurie, etc); or having an appropriate analytic language to speak of objects of higher topoi (and then, Grothendieck's homotopy hypothesis would give us such a thing). Of course, the best thing would be to have both of them.